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Double Factorials: Selected Proofs and Notes Francis J. O’Brien, Jr.1 Aquidneck Indian Council Newport, RI USA November 20, 2009 ©2009 ₪ Table of Contents Introduction and Notation…………………………………………………................................... 1.0 Definitions………………………………………….…….………………………………….. 1.1 Examples…………………………………………………………………………….…... 2.0 Elementary Results……………………………...................................................................... 2.1 Positive Even Integers………………………………………………………….………... 2.1.1 Derivation……………………………………………………………….………… 2.2 Positive Odd Integers……………………………...….……………………..……..……. 2.2.1 Derivation…………………….…………………………………………………… 2.3 Partial Double Factorials………………………………………………………………… 2.4 Negative Odd Integers…………………………………………...……………………… 2.5 Fractional Double Factorials…………………………………………………………….. 2.6 Sums, Products, Rational Functions and Numbers……………………………………… 2.7 Miscellaneous Calculations……………………………….……....................................... 2.7.1 Fractional Factorials and Gamma Function ………………………………………. 2.7.2 Derivative of x!! ……………...………………………............................................ 3.0 Applications………………………………………………………………………………….. 4.0 Summary…………………………………………………….……………………………….. References………………………………………………………………………………………... Appendix………………………………………………………………........................................ 2 3 3 4 4 5 6 6 8 11 12 18 18 18 23 24 25 31 34 List of Figures Figure 1 Plots of Double Factorial Function and Gamma Function………………………… 3 List of Tables Table 1 Table 2 Table 3 Table 4 Examples of Common Positive Integer Double Factorial Expressions…………….. 7 Other Forms for n!! ………………………………………………………………... 14 Example of Writing Factorial Backwards ………………………………………….. 22 Summary of Generalized Functional Relations for Double Factorial Expressions.... 26 Copyright © 2009 by Francis Joseph O’Brien, Jr., Aquidneck Indian Council. All rights reserved. I thank Aimee Ross for assistance in carrying out some of the calculations. Ms. Ross is an undergraduate at the University of Massachusetts (Dartmouth), Department of Mathematics, and a Student Intern at Naval Undersea Warfare Center, Newport, RI. 1 Introduction Double factorials—especially for integers—are used widely in the disciplines of mathematics, pure & applied science, and engineering. The referenced double factorial formulas are selected from the listing in the Wolfram Internet website, http://functions.wolfram.com/GammaBetaErf/Factorial2/08/0002/. The Wolfram compendium provides an historical overview and interrelates the factorial, double factorial, Pochhammer symbol, binomial and multinomial coefficients. Here we propose new formulas. The gamma function, Γ( x ) , plays a prominent role in double factorial calculations. Artin’s brief book is the classic introduction to the complete gamma function (for real variables). Gradshteyn & Ryzhik is a standard reference work. Whittaker is good for derivations; the 1902 edition by Whittaker is available online through Google Books. Other sources are listed in References. A brief online summary is available at http://www.docstoc.com/docs/3507375/500Integrals-of-Elementary-and-Special-Functions. The derivations are elementary with many details for the benefit of students. Additional material will be added as it becomes available. I would appreciate receiving any reports of errors. Computational issues are not addressed in this report. See The National Institute of Standards and Technology, “Guide to Available Mathematical Software,” at http://gams.nist.gov/. Notation • • • • • • • • • arg is argument of a function cos is cosine function csc is cosecant function Γ( x ) is the (complete) gamma function int. means integer part of mixed fraction, ⎛ p⎞ p , p ≠ q ; int ⎜ ⎟ = 0, p < q. ⎜q⎟ q ⎝ ⎠ • • ln is natural logarithmic function n, x, z are variables, either integer or non-integer n ∈ N means all natural numbers, n = 0,1,2,3,K n ∈ N + means all positive natural numbers, n = 1,2,3,K ⎧the number, π = 3.141592636 L π is ⎨ o ⎩the number in radians, π = 180 (degrees) Product operator, ∏ k =m n f (k ) = f (m) f (m + 1) L f (n). If n < m, we define ∏ f(k) = 1 k =m n • • • • • • Ψ ( x ) is the Psi function (Gradsteyn & Rhyzik, Section 8.36) Re means real variable sin is sine function x! is factorial function x!! is double factorial function x!!! is triple factorial function © Francis J. O’Brien, Jr. <> 11/20/2009 Created on 1/23/2009 8:38:00 1.0 Definitions, Positive Integers 1.1 Examples ⎧2 ⋅ 4 ⋅ 6 L (n − 4 )(n − 2)n for n > 0, even ⎪ n!!= ⎨1 ⋅ 3 ⋅ 5L (n − 4 )(n − 2)n for n > 0, odd ⎪1 for n = 0,−1 ⎩ (2n + 1)!!= 1 ⋅ 3 ⋅ 5L (2n − 3)(2n − 1)(2n + 1) for n ≥ 0 (2n − 1)!!= 1 ⋅ 3 ⋅ 5L (2n − 5)(2n − 3)(2n − 1) for n ≥ 1 (2n )!!= 2 ⋅ 4 ⋅ 6L (2n − 4)(2n − 2)(2n) for n > 0 (2n + 2)!!= 2 ⋅ 4 ⋅ 6L (2n − 2)(2n)(2n + 2) for n ≥ 0 Note that, by definition, 0!!= −1!!= 1 NOTE: Partial product series and negative integers defined in Sects. 2.3 & 2.4. NOTE: a double factorial for integers skips by 2 only. A factorial that skips by three is called a triple factorial as in (3n)!!! for integer n. See Wikipedia article, http://en.wikipedia.org/wiki/Factorial for other definitions. Background and all basic formulas can be found at Wolfram Functions website: • http://functions.wolfram.com/GammaBetaErf/Factorial2/08/0002/ • COMMENT: A definition for “partial double factorial” series for products such as 6 ⋅ 8 ⋅ 10L n or 7 ⋅ 9 ⋅ 11L n is considered in Sect. 2.3. NOTE: a plot of the double factorial function and gamma function ▼generated by http://www.quickmath.com/. It is apparent that negative double factorial − x!! is not defined for even negative integers while negative odd integers, all positive integers and positive/negative fractions ± calculated. Representative double factorial values are displayed in table following Fig. 1. p !!, p ≠ q, can be q x Figure 1. Plots of Double Factorial Function and Gamma Function (Real Vari